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From Wikipedia
In mathematics, the polar coordinate system is a twodimensionalcoordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.
The fixed point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole with the fixed direction is the polar axis. The distance from the pole is called the radial coordinate or radius, and the angle is the angular coordinate, polar angle, or azimuth.
History
The concepts of angle and radius were already used by ancient peoples of the 1st millennium BCE. The Greek astronomer and astrologerHipparchus (190120 BCE) created a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions. In On Spirals,Archimedes describes the Archimedean spiral, a function whose radius depends on the angle. The Greek work, however, did not extend to a full coordinate system.
From the 8th century CE onward, Muslim astronomers developed methods for approximating and calculating the direction to Makkah (qibla)—and its distance—from any location on the Earth. From the 9th century onward they were using spherical trigonometry and map projection methods to determine these quantities accurately. The calculation is essentially the conversion of the equatorial polar coordinates of Mecca (i.e. its longitude and latitude) to its polar coordinates (i.e. its qibla and distance) relative to a system whose reference meridian is the great circle through the given location and the Earth's poles, and whose polar axis is the line through the location and its antipodal point.
The Persian geographer, AbÅ« RayhÄ�n BÄ«rÅ«nÄ« (9731048), developed ideas which are seen as an anticipation of the polar coordinate system. Around 1025 CE, he was the first to describe a polar equiazimuthal equidistant projection of the celestial sphere.
There are various accounts of the introduction of polar coordinates as part of a formal coordinate system. The full history of the subject is described in Harvard professor Julian Lowell Coolidge's Origin of Polar Coordinates.GrÃ©goire de SaintVincent and Bonaventura Cavalieri independently introduced the concepts in the midseventeenth century. SaintVincent wrote about them privately in 1625 and published his work in 1647, while Cavalieri published his in 1635 with a corrected version appearing in 1653. Cavalieri first used polar coordinates to solve a problem relating to the area within an Archimedean spiral. Blaise Pascal subsequently used polar coordinates to calculate the length of parabolic arcs.
In Method of Fluxions(written 1671, published 1736), SirIsaac Newton examined the transformations between polar coordinates, which he referred to as the "Seventh Manner; For Spirals", and nine other coordinate systems. In the journal Acta Eruditorum(1691),Jacob Bernoulli used a system with a point on a line, called the pole and polar axis respectively. Coordinates were specified by the distance from the pole and the angle from the polar axis. Bernoulli's work extended to finding the radius of curvature of curves expressed in these coordinates.
The actual term polar coordinates has been attributed to Gregorio Fontana and was used by 18thcentury Italian writers. The term appeared in English in George Peacock's 1816 translation of Lacroix's Differential and Integral Calculus. Alexis Clairaut was the first to think of polar coordinates in three dimensions, and Leonhard Euler was the first to actually develop them.
Common conventions
The radial coordinate is often denoted by r, and the angular coordinate by Î¸ or t.
Angles in polar notation are generally expressed in either degrees or open source program that can generate two dimensional plots of mathematical functions and data sets.
Features
Graph supports entering functions on regular cartesian as well as parametric and polar form. Functions can be traced and at a given coordinate the function value and the two first derivatives is displayed. Graph is available in 23 different languages. A linux version isn't planned, yet the program works well using WINE.
From Yahoo Answers
Answers:I know that you can with maple. You can download a trial version of it, but the nontrial is not free. It is not the easiest version to learn, but it is compatible with word (can copy and paste into word easily). Also if you already have a TI83 you could use the cord to hook into the computer. I don't know if you can put the graphs into word, but it is worth a try.
Answers:I agree with what you are doing, but do not follow your reason. Here is how I explain it: As t goes from 0 to pi the end of radius r draws out a curve above the horizontal axis. Because the curve does not intersect itself, and because it lies entirely above the horizontal axis (endpoints on the axis), it has area bounded above by the curve and below by the horizontal axis. Call that area Aup. As t goes from pi to 2 pi, the mirror image of the curve is drawn entirely below the horizontal axis, and that part has of the curve has area bounded by the lower curve and the horizontal axis, Alower. Because of the mirror image symmetry, you can calculate the total area either by a full integration over 0 to 2 pi, Afull, or you can calculate the area of the upper half and double it, Afull = 2 Aup ( = 2 Alower). Area full = Integral of r(t) from 0 to 2 pi Areaup = Integral of r(t) from 0 to pi Areafull = 2 * Areaup I do not believe saying the 2 above is canceling out the 1/2 in the expression 1/2 r^2 dt. In effect one cancels the other, but that is just a coincidence, and not a reason for the method. The reason is the symmetry. If a function r(t) did not have symmetry above and below the horizontal axis, you could not do this. (Example: r = 3[1+cos(t/3))
Answers:the input is r = f( ) It is for plotting functions, not individual points, you could enter r1 = 1 + 2 to get an Archimedean spiral
Answers:You need to use either cylindrical coordinates or spherical coordinates. Both of them start with polar coordinates and extend them in different ways to 3d.
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